Uniformly recurrent subgroups and the ideal structure of reduced crossed products

We study the ideal structure of reduced crossed product of topological dynamical systems of a countable discrete group. More concretely, for a compact Hausdorff space $X$ with an action of a countable discrete group $\Gamma$, we consider the absence of a non-zero ideals in the reduced crossed product $C(X) \rtimes_r \Gamma$ which has a zero intersection with $C(X)$. We characterize this condition by a property for amenable subgroups of the stabilizer subgroups of $X$ in terms of the Chabauty space of $\Gamma$. This generalizes Kennedy's algebraic characterization of the simplicity for a reduced group $\mathrm{C}^{*}$-algebra of a countable discrete group.